1 Homework set #12 Phy 3031 Modern physics, Spring 2007 Not to be handed in on Wednesday, April 4, 2007. For numerical answers give two significant digits. The first should be correct, the second can be approximate. 1. In a region of space, a particle has a wave function given by ψ(x) = Ae−x energy E = h̄2 /2mL2 , where L is some length. (a) (b) (c) (d) (e) 2 /2L2 and Find the potential V (x) as a function of x. What classical object has a potential energy function of this form? Find the kinetic energy as a function as a function of position x. What is the value of x for the classical turning point of this particle? The potential energy of a simple harmonic oscillator in terms of its angular frequency ω is given by V (x) = 12 mΩ2 x2 . Compare this with your answer to part (a), and show that the total energy for this wave function can be written as E = 12 h̄Ω. 2. A particle of mass m is confined to move in only one dimension over a range from x = 0 to x = L. (a) Use the uncertainty relationship to estimate the smallest possible energy. (b) Assume that there is no force on the particle between the two ends of the region. Use the Schrödinger equation to find the ground state energy for the particle. (c) Are your answers to part (a) and part (b) close to each other? 3. An electron is trapped in a one-dimensional infinite square well in the n = 5 state. (a) Find the probability of finding the electron between x = 0.2L and x = 0.4L. (b) Find the probability of finding the electron between x = L/2 − 0.005L and x = L/2 + 0.005L. 4. For a particle of mass m in the n = 3 state of an infinite square well from x = 0 to x=L (a) find hxi, the average value of x. This is also known as the “expectation value” of x. (b) For the same state, also find hx2 i, the average value of x2 . This is also known as the “expectation value” of x2 .